Representations of Reductive Groups

Over four decades David Vogan’s groundbreaking work in representation theory has changed the face of the subject. We give a brief summary here.

Exotic sheaves are certain complexes of coherent sheaves on the cotangent bundle of the flag variety of a reductive group. They are closely related to perverse-coherent sheaves on the nilpotent cone. This expository article includes the definitions of these two categories, applications, and some structure theory, as well as detailed calculations for SL₂.

The main result of [4] is the description of an algorithm to compute the signature of the Hermitian form on an irreducible representation of a real reductive Lie group G, and therefore determine if it is unitary. This paper concerns an important ingredient of the algorithm. If the inner class of G is defined by an outer automorphism δ, so that G does not have discrete series representations, it is necessary to compute a new class of Kazhdan–Lusztig–Vogan polynomials for G. These were defined and studied by Lusztig and Vogan in [10]. In order to carry out the computation, we introduce new class of twisted parameters, and study the Hecke algebra action in the resulting basis.

In this paper, we recover certain known results about the ladder representations of \(\mathrm{GL}(n, \mathbb{Q}_{p})\) defined and studied by Lapid, Mínguez, and Tadić. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa–Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal formula proved by Lapid–Mínguez and Tadić is a direct consequence of the BGG resolution of finite-dimensional simple \(\mathfrak{g}\mathfrak{l}(n)\)-modules. We make a connection between the semisimplicity of Hecke algebra modules, unitarity with respect to a certain hermitian form, and ladder representations.

Let k be a field, let G be a reductive group, and let V be a linear representation of G. Let \(V/\!/G =\mathop{ \mathrm{Spec}}\nolimits ({\mathrm{Sym}}^{{\ast}}(V ^{{\ast}}))^{G}\) denote the geometric quotient and let \(\pi: V \rightarrow V/\!/G\) denote the quotient map. Arithmetic invariant theory studies the map π on the level of k-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory”, we provide necessary and sufficient conditions for a rational element of \(V/\!\!/G\) to lie in the image of π, assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.

Spiral Schubert varieties are conjecturally the only Schubert varieties in type \(\widetilde{A}_{2}\) for which rational smoothness at a torus-fixed point is not detected by the number of torus-invariant curves passing through that point. In this paper we determine the locus of smooth points of a spiral Schubert variety of type \(\widetilde{A}_{2}\). This continues the study begun in [7], where the locus of rationally smooth points was determined. The main result describes the smooth locus in terms of the action of the Weyl group on \(\mathbb{R}^{2}\); using this result, we identify the maximal singular points of these varieties. We make key use of the results of [7] relating the Bruhat order to the Weyl group action on \(\mathbb{R}^{2}\).

In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups.

The first part (Sections 2–7) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with \((\mathfrak{g},K)\)-cohomology and nilpotent Lie algebra cohomology; the second part (Sections 8–13) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G′ (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are a priori known to be “nice” in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

This paper connects results on Amitsur–Levitski identities for simple Lie algebras, ideals in Borel subalgebras, commutative Lie subalgebras in simple Lie algebras, filtration of sheets, and recent work with Nolan Wallach on the variety of singular elements in a complex semisimple Lie algebra.

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed purely in terms of the Weyl group and the indexing set is independent of the characteristic.

Let W be a Coxeter group and let M be the free Z[v, v ⁻¹]-module with basis indexed by the involutions of W. We show how the recent results of Elias and Williamson on Soergel bimodules can be used to give an alternative definition of an action of the Hecke algebra of W on M​.

The Iwahori–Hecke algebra \(\mathcal{H}\) of a Coxeter system (W, S) has a “standard basis” indexed by the elements of W and a “bar involution” given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan–Lusztig basis of \(\mathcal{H}\) is a canonical basis. Lusztig and Vogan defined a representation of a modified Iwahori–Hecke algebra on the free \(\mathbb{Z}[v,v^{-1}]\)-module generated by the set of twisted involutions in W, and showed that this module has a unique pre-canonical structure compatible with the \(\mathcal{H}\)-module structure, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan–Lusztig basis. One can modify the definition of Lusztig and Vogan’s module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this manner, and explain the relationships between their resulting canonical bases. Some of these canonical bases are equivalent in a trivial fashion to Lusztig and Vogan’s construction, while others appear to be unrelated. Along the way, we also clarify the differences between Webster’s notion of a canonical basis and the related concepts of an IC basis and a P-kernel.

We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of \(\mathrm{SL}(2, \mathbb{R})\).

This paper is concerned with the structure of packets of representations and some refinements that are helpful in endoscopic transfer for real groups. It includes results on the structure and transfer of packets of limits of discrete series representations. It also reinterprets the Adams–Johnson transfer of certain nontempered representations via spectral analogues of the Langlands–Shelstad factors, thereby providing structure and transfer compatible with the associated transfer of orbital integrals. The results come from two simple tools introduced here. The first concerns a family of splittings of the algebraic group G under consideration; such a splitting is based on a fundamental maximal torus of G rather than a maximally split maximal torus. The second concerns a family of Levi groups attached to the dual data of a Langlands or an Arthur parameter for the group G. The introduced splittings provide explicit realizations of these Levi groups. The tools also apply to maps on stable conjugacy classes associated with the transfer of orbital integrals. In particular, they allow for a simpler version of the definitions of Kottwitz–Shelstad for twisted endoscopic transfer in certain critical cases. The paper prepares for spectral factors in twisted endoscopic transfer that are compatible in a certain sense with the standard factors discussed here. This compatibility is needed for Arthur’s global theory. The twisted factors themselves will be defined in a separate paper.

Lower bounds to the Gelfand–Kirillov dimension of discrete series are given for semisimple Lie groups with finite center by showing that the K-finite vectors are torsion free with respect to enveloping algebras of certain unipotent subgroups. In particular we prove two folk theorems about the Gelfand–Kirillov dimension. The first is that the holomorphic (or anti-holomorphic) discrete series are the “smallest” and representations with Whittaker models for minimal parabolic subgroups are the “largest” (a more precise result in the quasi-split case is due to Kostant). We also show that if G is quaternionic and not of type A or C, then the quaternionic discrete series is the “smallest”.

We show that simple highest weight modules for \( \mathfrak{s}\mathfrak{l}_{12}(\mathbb{C}) \) may have reducible characteristic variety. This answers a question of Borho–Brylinski and Joseph from 1984. The relevant singularity under Beilinson–Bernstein localization is the (in)famous Kashiwara–Saito singularity. We sketch the rather indirect route via the p-canonical basis, W-graphs and decomposition numbers for perverse sheaves that led us to examine this singularity.